traditional time series analysis tries to divide time series into a limited number of components, each one a function of time \(t\)
trend
season
other cyclic components
noise
noise is what we consider the remaining variability that cannot be explained by the other components
Components of time series data
there is not the one correct decomposition
the choice of components and how we try to model them depends on what we know about the process
say we know that daily mean temperature (at a certain location) follows a seasonal periodicity, so we could try to model this component with a periodic function of time \(f(t)\), like a sinus function
there could also be another (overlaying) trend, e.g., for temperature a long term increase (climate change)
the following slide shows a similar example using CO2 measurements
Components of time series data (here CO2)
Code
library(graphics) #load package including datadata(co2) # load datam=decompose(co2) plot(m)
Components of time series data
in this course we will build on a basic additive component model
\[y(t) = m(t) + e(t)\]
\(m(t)\) is the deterministic component, which we can explain by, e.g., a constant increase (linear trend), a periodic function, a combination of both, …
\(e(t)\) is the stochastic component, that may or may not be random (white noise)
cannot be explained by the deterministic functions we know (in the context of this course)
can have different forms, can show patterns
might be modelled by, e.g., AR(p) or MA(q) processes
Back to the temperature series
Removing the diurnal periodicity
Assuming this is a sinus function, \[\alpha_1 + \alpha_2 \sin(t + \alpha_3)\], we need non-linear regression (function is non-linear in \(\alpha_3\))
attach(meteo) # so we can use variable names directly:f =function(x) sum((T.outside - (x[1]+x[2]*sin(pi*(hours+x[3])/12)))^2)nlm(f,c(0,0,0))
# plot data and trend:plot(T.outside~date,meteo,type='l')meteo$T.per =18.2-4.9*sin(pi*(meteo$hours+1.6)/12) #use parameters fitted before (previous slide)lines(T.per~date,meteo,col='red')
A possible approach is to find the minimum for Akaike’s Information Criterion (AIC) for ARMA(\(p,q\)) models and series of length \(n\): \[AIC = \log \hat{\sigma}^2 + 2(p+q+1)/n\] with \(\hat{\sigma}^2\) the estimated residual (noise) variance.
Instead of finding a single best model using this single criterion, it may be better to select a small group of “best” models, and look at model diagnostics for each: is the residual white noise? does it have stationary variance?
Even better may be to keep a number of “fit” models and consider each as (equally?) suitable candidates.
Prediction of Anomaly, e.g. with AR(6) - 110 minutes
Code
pltpred2(c(9400, 10000), "110 minutes")
Prediction, e.g. with AR(6) - 1 day
Code
pltpred(c(8000, 11330), "predicting 1 day")
Prediction of Anomaly, e.g. with AR(6) - 1 day
Code
pltpred2(c(8000, 11330), "predicting 1 day")
Prediction, e.g. with AR(6) - 1 week
Code
pltpred(c(1, 19970), "predicting 1 week")
Prediction of Anomaly, e.g. with AR(6) - 1 week
Code
pltpred2(c(1, 19970), "predicting 1 week")
Predicting 1 week: periodic trend + ar(6) for anomaly
Now that we have an idea of the trend and the anomaly, we can combine them to make predictions
Predicting 1 week: periodic trend + ar(6) for anomaly
Code
plot(T.outside,xlim=c(1,19970), type='l')x.an =arima(an, c(6,0,0)) # model the anomaly by AR(6)x.pr =as.numeric(predict(x.an, 10080)$pred) x.se =as.numeric(predict(x.an, 10080)$se)hours.all =c(meteo$hours, max(meteo$hours) + (1:10080)/60)T.per =18.2-4.9*sin(pi*(hours.all+1.6)/12)lines(T.per, col ='blue')hours.pr =c(max(meteo$hours) + (1:10080)/60)T.pr =18.2-4.9*sin(pi*(hours.pr+1.6)/12)lines(9891:19970, T.pr+x.pr, col='red')lines(9891:19970, T.pr+x.pr+2*x.se, col='green')lines(9891:19970, T.pr+x.pr-2*x.se, col='green')title("predicting 1 week: periodic trend + ar(6) for anomaly")
Simulation with and without trend
We can also use these models to simulate data
Code
x =arima(T.outside, c(6,0,0))plot(T.pr +arima.sim(list(ar = x.an$coef[1:6]), 10080, sd =sqrt(0.002556)), col ='red', ylab="Temperature")lines(18+arima.sim(list(ar = x$coef[1:6]), 10080, sd=sqrt(0.002589)), col ='blue')title("red: with trend, blue: without trend")
What can we learn from this?
Prediction/forecasting:
AR(6) prediction is a compromise between the end of the series and the trend
the closer we are to observations, the more similar the prediction is to the nearest (last) observation
further in the future the prediction converges to the trend
the more useful (realistic) the trend is, the more realistic the far-into-the-future prediction becomes
the standard error of prediction increases when predictions are further in the future.
A side note: fitting a phase with a linear model
A phase shift model \(\alpha \sin(x + \phi)\) can also be modelled by \(\alpha sin(x) + \beta cos(x)\); this is essentially a re-parameterization.
A side note: fitting a phase with a linear model
A phase shift model \(\alpha \sin(x + \phi)\) can also be modelled by \(\alpha sin(x) + \beta cos(x)\); this is essentially a re-parameterization.
Try the following code:
Code
x =seq(0, 4*pi, length.out =200) # x plotting rangesincos =function(dir, x) { # plot the combination of a sin+cos function, based on dir a =sin(dir) b =cos(dir)# three curves:plot(a *sin(x) + b *cos(x) ~ x, type ='l', asp=1, col ='green')lines(x, a *sin(x), col ='red')lines(x, b *cos(x), col ='blue')# legend:lines(c(10, 10+a), c(2,2), col ='red')lines(c(10, 10), c(2,2+b), col ='blue')arrows(x0 =10, x1 =10+a, y0 =2, y1 =2+b, .1, col ='green')title("red: sin(x), blue: cos(x), green: sum")}for (d inseq(0, 2*pi, length.out =100)) {sincos(d, x)Sys.sleep(.1)}
A side note: fitting a phase with a linear model
So, we can fit the same model by a different parameterization:
we have now seen how to select models and fit them to the data using R-functions like nlm()
in the next lectures, we will have a look at different methods that are used for this fitting (optimization)
Sidetrack: time series analysis for satellite-based land use change detection
Satellites continusouly observe the surface of the earth
How can we use this information to detect and monitor changes like deforestration?
Example dataset: MODIS 16 day vegetation index measaurements over the Brazilian amazon
A time-series of images
The EVI (enhanced vegetation index) is a one-dimensional index computed from near infrared, red, and blue reflection measurements. Values can be interpreted as “greenness” and range from -1 to 1.
A time-series of images
For each of the \(1200^2\) pixels, we interpret the set of observations as time series.
Consider the general model: \[y_t = T_t + S_t + R_t\]
\(T_t\) describes the trend
\(S_t\) describes seasonal effects
\(R_t\) describes remaining variation
We could be interested in both, changes in the trend and seasonality.
Example analysis
An example series (1 pixel in forest area):
Example analysis
We compute differences from the overall mean to center the series around 0.
Example analysis
We compute differences from the overall mean to center the series around 0.
Example analysis
We assume yearly seasonality and build seasonal sub-series:
\[
\begin{aligned}
y_t^{(16)} &= (y_{2005,16},y_{2006,16},y_{2007,16},y_{2008,16}, ...) \\
y_t^{(32)} &= (y_{2005,32},y_{2006,32},y_{2007,32},y_{2008,32}, ...) \\
...
\end{aligned}
\] for which we simply compute mean values.
Example analysis
After subtracting the seasonal component, the series looks like:
Example analysis
After removing large parts of seasonality, we can study the trend of the data. Deforestation should be visible by abrupt changes in the trend structure. To test, whether there is a structural change, we assume the following linear regression model \[T_t = \beta_0 + \beta_1 t\] and hypothesize that the regression coefficients \(\beta_0, \beta_1\) do not change in time.
Example analysis
Fitting a global trend leads to:
Example analysis
Looking at the cumulated sum of the model’s residuals shows that they do not fluctuate around 0, which we expected in our hypothesis.
By thresholding the cumulated residuals based on a given confidence level, we can find a breakpoint around t=107.
Example analysis
A better model:
Example analysis
Literature (sidetrack)
Verbesselt, J., Hyndman, R., Newnham, G., & Culvenor, D. (2010). Detecting trend and seasonal changes in satellite image time series. Remote Sensing of Environment, 114, 106-115. DOI: 10.1016/j.rse.2009.08.014.
R. B. Cleveland, W. S. Cleveland, J.E. McRae, and I. Terpenning (1990) STL: A Seasonal-Trend Decomposition Procedure Based on Loess. Journal of Official Statistics, 6, 3-73.
W. S. Cleveland, E. Grosse and W. M. Shyu (1992) Local regression models. Chapter 8 of Statistical Models in S eds J.M. Chambers and T.J. Hastie, Wadsworth and Brooks/Cole.
